The below answer is not actually accurate. There is an established appeals procedure for denial of a civil service position in New York City, including as an NYPD officer. We handle these appeals all the time.
This is not really a criminal issue. Unfortunately the NYPD can reject you for any non-discriminatory reason. My guess is this was for a failure to reveal the arrest. If you contact my office on Monday, I might be able to help.
This isn't really a criminal defense question. If you weren't legally arrested the NYPD shouldn't be able to hold it against you even though they have a lot of discretion. You may want to reach out to an employment attorney and/pr repost this question in the employment section.